Dualities in the theory of accessible categories

TL;DR

This paper unifies and extends various dualities in accessible category theory through a general duality theorem based on weakly sound classes of weights, including new dualities in the enriched setting.

Contribution

It introduces a unifying duality theorem for accessible categories using weakly sound classes of weights, extending known dualities and discovering new ones in the enriched context.

Findings

Unified framework for known dualities

Extension to enriched categories

Discovery of new dualities

Abstract

Through the notion of weakly sound class of weights, we recover many known dualities involving accessible categories with a chosen class of limits, as instances of a general duality theorem. These include the Gabriel-Ulmer duality for locally finitely presentable categories, Diers duality for locally finitely multipresentable categories, and the Makkai-Par\'e duality for finitely accessible categories. In doing so, we extend these to the enriched setting, provide a more formal and unifying approach to the theory, and also discuss new dualities that arise as a consequence of our main theorem.